Motional resonances of three-dimensional dual-species Coulomb crystals
Byoung-moo Ann, Fabian Schmid, Jonas Krause, Theodor W. Hänsch, Thomas Udem, Akira Ozawa
MMotional resonances of three-dimensionaldual-species Coulomb crystals
Byoung-moo Ann , Fabian Schmid , Jonas Krause, Theodor W Hänsch,Thomas Udem and Akira Ozawa Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, GermanyE-mail: [email protected] and [email protected] 31 October 2018, revised 28 November 2018Accepted for publication 4 December 2018Published 4 January 2019
Abstract
We investigate the motional resonances of dual-species Coulomb crystals comprised of Be + and Mg + ions held in a 4-rod linear Paul trap. Our experimental data and simulations show that thesecular motion of such mixed crystals has rich dynamics. Their secular spectra can differsigni fi cantly from those of pure ion crystals. We propose a simple model based on mechanicalcoupling with Coulomb interactions between the two different ion species that explains manyfeatures of the secular spectrum. Our fi ndings contribute to a more reliable identi fi cation of theion species in mixed crystals.Keywords: ion traps, Coulomb crystals, secular motion
1. Introduction
Laser cooling of atoms [ ] and trapped ions [ ] has allowed tosuppress systematic effects such as Doppler broadening andshifts and has set the stage for today ’ s optical precisionmetrology. Laser cooling techniques nowadays even make itpossible to cool down to μ K or nK temperatures [ – ] and insome cases to the motional ground state [
6, 7 ] . This devel-opment was accompanied with the emergence of highly stablelaser systems [ – ] to probe narrow-band transitions. Sym-pathetic cooling provides a way to cool molecular or atomicion species that do not possess suitable cooling transitions [ – ] . In this scheme the sympathetically cooled ions aremixed with coolant ions which have a convenient coolingtransition. Coulomb interaction between the ions then leads tothermalization of the whole ensemble. With suf fi cient coolingpower, the two ion species eventually form mixed Coulombcrystal structures that have been described in previous pub-lications [ –
17, 21 – ] .While the coolant ions can be easily located by observingtheir fl uorescence induced by the cooling laser, it is moredif fi cult to detect the presence of the sympathetically cooled ions. Observation of dark spots within the Coulomb crystal isone method, but this is not very quantitative in large crystals,and offers little or no information on the type of ion. Excitingthe secular motional resonances is another widespreadmethod [
16, 24, 25 ] . In this case the motion is coupled to thecoolant ions which can be observed optically [
26, 27 ] . Thesecular motion leads to Doppler broadening of the coolingtransition which typically results in an increased or decreased fl uorescence rate when the cooling laser is far red detuned ornear resonance respectively.In contrast to a large Coulomb crystal, the secular motionof a single trapped and laser cooled ion is simple and can bedescribed as a damped harmonic oscillator. The strength ofthe con fi nement is given by the trap voltages and the charge-to-mass ratio of the ion. The con fi nement force, the ion mass,and the dissipative force induced by the cooling laser deter-mine the frequency and the quality of the resonance. Oncemore than one ion form a Coulomb crystal, their mutualrepulsion gives rise to additional forces. Therefore, theirmotional resonances are different from those of the individualions . It is of importance to model such mechanical couplingbetween the ion species to better understand the secular Journal of Physics B: Atomic, Molecular and Optical PhysicsJ. Phys. B: At. Mol. Opt. Phys. ( ) ( ) https: // doi.org / / / aaf5ea Present Address: Kavli Institute of Nanoscience, Delft University ofTechnology, Lorentzweg 1, 2628 CJ Delft, The Netherlands. One exception is the center-of-mass motion of the ion crystal, where thetotal mass-to-charge ratio still uniquely determines the secular frequency. / / + Made open access 11 January 2019
Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. pectra of mixed ion crystals. The longitudinal vibrationmodes of a linear chain of identical ions can be readilyobtained, with up to 3 ions even in analytical form [ ] . Thismodel may be readily extended to mixed ions and radialvibrations. Experimentally, a linear chain is obtained withnot too many ions stored in a linear Paul trap and / or withstrong radial con fi nement. Mixed crystals that consist of afew ions fi nd important applications in quantum informationprocessing and precision spectroscopy [
13, 29, 30 ] . Whenthe number of ions exceeds a certain threshold or the axialcon fi nement is strong, a three-dimensional structure isformed [ ] . Mixing two species in this case leads to aseparation of the species [
11, 12 ] . Ions with a smaller mass-to-charge ratio experience stronger con fi nement and form a ‘ core ’ surrounded by a ‘ shell ’ of heavier ions. In the case ofa three-dimensional structure consisting of a large number ofions, it is no longer feasible to obtain exact analyticalsolutions for the secular spectra. A simple explanation of themechanical coupling between the two ion species is given byRoth et al [ ] and Zhang et al [
17, 22 ] . In this model, thepotential induced by the core or shell ions signi fi cantlychanges the potential of the other species. The dynamics ofsympathetically cooled ions under the in fl uence of the spacecharge of laser cooled ions are also modeled by Baba and Waki [ ] .Even though it has been shown that the secular motion ofmixed ion crystals behaves differently from that of pure ioncrystals, there was still no full understanding how the two ionspecies mechanically interact and lead to a modi fi ed secularspectrum. In this work, we present an intuitive model thatdescribes the coupling between two ion species due to theCoulomb interaction. Based on this model, we fi nd that thesecular spectra of ion crystals with a ‘ core ’ and ‘ shell ’ structureexhibit characteristic features over a wide range of crystalparameters. The resonance of the lighter ion species is broa-dened and shifted to higher frequencies, while the resonance ofthe heavier species is split into two components. Our model,which partially is an extension of the space charge modelproposed in [ ] , explains the origin of these effects. Themodel is con fi rmed both by experiment and by MD simulation,where the secular spectra of the individual ions are investi-gated. Most experimental studies of two-species trapped ioncrystals have been performed in the limit of very few ions [
13, 29, 30 ] , or in situations where only one of the species ( thecoolant ) can be observed directly via laser induced fl uores-cence imaging [ –
16, 19, 23 ] . Here, we report on experimentswhere both ion species, Be + and Mg + , permit direct laserimaging, so that experiments and molecular dynamics simu-lations can be readily compared. The paper is organized asfollows. In section 2 we study the motional properties of themixed ion crystals by employing MD simulations. Severalinteresting features are highlighted, and a simple model ispresented to explain them. Experimental results are given insection 3. The last section summarizes our fi ndings.
2. Theoretical description
MD simulations are widely used to investigate the motion ofatoms and molecules [ ] . We use this method to model thedynamics of our mixed Be + and Mg + Coulomb crystals. Wedeveloped an MD software written in C ++ which uses the fi fth-order Runge – Kutta method with adaptive time steps forsolving the multi-particle equations of motion.As sketched in fi gure 1 ( a ) , our trap is a 4-rod linear Paultrap with ring electrodes for axial con fi nement [ ] . To savecomputation time, in the MD simulation, the time-averagedsecular potential is employed x y z e Vm r x y eUz z x y , , 4 12 .1 k F = W + + - + ⎜ ⎟ ⎛⎝ ⎞⎠ ( ) ( ) ( )( )
Here, V = Ω = π × ( RF ) applied to the rod elec-trodes, while U =
265 V is the dc voltage applied to the ringelectrodes. The distance between the two ring electrodes is z = with κ = r = ( ) ignores possible effects due to the micromo-tion. The approximation is justi fi ed because the time scale ofthe secular motion is an order of magnitude slower than thatof the micromotion. This is con fi rmed by comparing theresults of some simulations using the full time-dependentpotential with results obtained using the time-averagedsecular potential ( see appendix A ) .To model the laser cooling, we assume an isotropiccontinuous drag force f v drag a = - that acts on both ionspecies. In the experiments we also laser cool both ionspecies. The linear dependence on velocity v is a goodapproximation when the harmonic oscillations at the secularfrequencies ω j ( j = x , y , z ) lead to Doppler shifts that aremuch smaller than the line width of the cooling transition.The drag force model also assumes the weak bindingregime, that is, the secular frequencies are much smaller thanthe line width of the cooling transition. In this regime thereare many absorption and emission events per secular oscil-lation period such that a continuous drag force is a goodapproximation. This drag force leads to a vanishing temp-erature. In a real experiment random momentum kicks due tospontaneous emission and collisions with the backgroundgas would give rise to heating effects and lead to a fi nitetemperature even when laser cooling is engaged. We haverun some simulations that include heating [ ] and fi nd thata fi nite temperature of a few mK does not affect our quali-tative description of motional coupling of mixed ion crys-tals. Hence, we ignore these heating effects for the resultsobtained in this work. ( )035002 B-m Ann
265 V is the dc voltage applied to the ringelectrodes. The distance between the two ring electrodes is z = with κ = r = ( ) ignores possible effects due to the micromo-tion. The approximation is justi fi ed because the time scale ofthe secular motion is an order of magnitude slower than thatof the micromotion. This is con fi rmed by comparing theresults of some simulations using the full time-dependentpotential with results obtained using the time-averagedsecular potential ( see appendix A ) .To model the laser cooling, we assume an isotropiccontinuous drag force f v drag a = - that acts on both ionspecies. In the experiments we also laser cool both ionspecies. The linear dependence on velocity v is a goodapproximation when the harmonic oscillations at the secularfrequencies ω j ( j = x , y , z ) lead to Doppler shifts that aremuch smaller than the line width of the cooling transition.The drag force model also assumes the weak bindingregime, that is, the secular frequencies are much smaller thanthe line width of the cooling transition. In this regime thereare many absorption and emission events per secular oscil-lation period such that a continuous drag force is a goodapproximation. This drag force leads to a vanishing temp-erature. In a real experiment random momentum kicks due tospontaneous emission and collisions with the backgroundgas would give rise to heating effects and lead to a fi nitetemperature even when laser cooling is engaged. We haverun some simulations that include heating [ ] and fi nd thata fi nite temperature of a few mK does not affect our quali-tative description of motional coupling of mixed ion crys-tals. Hence, we ignore these heating effects for the resultsobtained in this work. ( )035002 B-m Ann et al s can be seen from equation ( ) , the radial trappingpotential is inversely proportional to the mass of the ions m .Therefore, lighter ions are more strongly bound to the trapcenter than heavier ions. Secular frequencies for a single Mg + ion are approximately ω z = π ×
115 kHz along theaxial direction and ω x , y = π ×
370 kHz along the radialdirection. For a single Be + ion the frequencies are ω z = π ×
187 kHz and ω x , y = π × Mg + experiences a radial restoring force of2.1 × − N at a distance of 1 μ m from the trap axis. Thedrag force constant α was set for both ion species to2.58 × − kg s – in our simulations. This value affects thecooling time and the spectral width of the vibrational reso-nances, but has little in fl uence on the secular frequencies.In the simulations the ions ’ initial positions are randomlychosen, and their initial velocities are set to zero. In the nextstep the ions are subjected to the trapping and cooling forcesuntil their positions reach a steady state. For the parametersgiven in this work, 3 ms is enough to crystallize about onehundred ions. An example of a crystal structure obtained inthis way is given in fi gure 1 ( b ) . As expected, the Mg + ionsform a shell structure surrounding the Be + ions that arelocated along the trap center. Once the ions are crystallized by the trap potential and thedrag force, we introduce a sinusoidal excitation force alongthe x direction f xf t cos ex 0 w = ˆ ( ) to induce motional excita-tion. Here, x ˆ is a unit vector along the x direction, f is theamplitude of the excitation force, and w p is the excitation frequency. In the experiment, the force is induced by mod-ulating the voltage of one of the outer electrodes ( see fi gure 1 ( a )) . Since the distance between the electrode and thetrap center is suf fi ciently large compared to the size of the ioncrystal, a uniform force is a good approximation. In principlethe ion motion is not exactly harmonic, but contains non-linearities due to the Coulomb interaction between the ions.These nonlinearities become negligible when the drivingforce is weak enough, such that the ion positions do notdeviate too much from their equilibrium positions. In thiswork, we focus on the fundamental frequencies and thereforeemploy only small amplitudes of the driving force by setting f = - N in the MD simulation. This is much smallerthan the typical radial restoring force of the trap. As a result,the non-harmonic component of the ion motion is negligiblein the simulation.After the excitation force is turned on at t =
0, the ionsstart oscillating with the frequency ω . The oscillation amplitude fi rst grows and after some time ( approximately 300 μ s in oursimulations ) stabilizes by balancing with the cooling force. Inthis state we determine the amplitude Max ( x i ( t )) – Min ( x i ( t )) and the mean position x i á ñ averaged over one oscillation period.The ions are numbered by the index i . We only consider themotion along the x direction since the motions along y and z aremuch smaller ( about 10% of the motion along x direction ) .From this data we determine the secular spectrum as theamplitude averaged over all Mg + or all Be + ions. An exampleis given in fi gure 2 together with the secular spectra of a singleBe + ion and a single Mg + ion for comparison. The Be + spectrum in the mixed crystal is signi fi cantly broadened com-pared with the single ion spectrum and the peak is found to be Figure 1. ( a ) Ion trap and geometry used in this work, radial ( left ) and axial view ( right ) . The cooling and photo-ionization beams for each ionspecies are directed along the trap axis ( z ) . The ion crystals are imaged on an EMCCD camera ( not shown ) looking perpendicular on the trapaxis. The outer electrodes depicted with dashed circles ( axial view only ) are for compensation of stray fi elds. An alternating voltage is appliedto one of the outer electrodes in order to excite secular motion of the ions. ( b ) A dual-species Coulomb crystal obtained by the moleculardynamics simulation. It consists of 80 Mg + ions ( light red spheres ) and 20 Be + ions ( dark blue spheres ) . Left: radial view; right: axial view ofthe same Coulomb crystal. ( )035002 B-m Ann
0, the ionsstart oscillating with the frequency ω . The oscillation amplitude fi rst grows and after some time ( approximately 300 μ s in oursimulations ) stabilizes by balancing with the cooling force. Inthis state we determine the amplitude Max ( x i ( t )) – Min ( x i ( t )) and the mean position x i á ñ averaged over one oscillation period.The ions are numbered by the index i . We only consider themotion along the x direction since the motions along y and z aremuch smaller ( about 10% of the motion along x direction ) .From this data we determine the secular spectrum as theamplitude averaged over all Mg + or all Be + ions. An exampleis given in fi gure 2 together with the secular spectra of a singleBe + ion and a single Mg + ion for comparison. The Be + spectrum in the mixed crystal is signi fi cantly broadened com-pared with the single ion spectrum and the peak is found to be Figure 1. ( a ) Ion trap and geometry used in this work, radial ( left ) and axial view ( right ) . The cooling and photo-ionization beams for each ionspecies are directed along the trap axis ( z ) . The ion crystals are imaged on an EMCCD camera ( not shown ) looking perpendicular on the trapaxis. The outer electrodes depicted with dashed circles ( axial view only ) are for compensation of stray fi elds. An alternating voltage is appliedto one of the outer electrodes in order to excite secular motion of the ions. ( b ) A dual-species Coulomb crystal obtained by the moleculardynamics simulation. It consists of 80 Mg + ions ( light red spheres ) and 20 Be + ions ( dark blue spheres ) . Left: radial view; right: axial view ofthe same Coulomb crystal. ( )035002 B-m Ann et al p-shifted ( by 18.5 kHz ) . A more dramatic change is observedfor Mg + . Its secular resonance is split into two components,one of which is down-shifted to 276 kHz, and the other is up-shifted to 408 kHz compared with the single ion resonance at370 kHz. + core ions To obtain a more detailed picture, we investigate the secularspectra of individual ions within the crystal. We fi rst have alook at the 20 Be + core ions of the crystal in fi gure 1 ( b ) .Apparently all these spectra are different as shown in fi gure 3.For the case of the core ions located near the center, thesecular resonance tends to be up-shifted, while the ionslocated at the ends experience a down-shift of their secularresonance. These effects yield an inhomogeneous broadeningof the averaged spectrum as shown in the bottom plot of fi gure 3. A qualitative explanation of the position dependencecan be given in following way: the Mg + ion shell generates apotential in addition to the trap potential . For the core ions inthe center of the trap, this leads to an enhancement of thecon fi nement and hence to a larger secular frequency. On theother hand, the core ions at the ends are not surrounded by therepelling shell ions and hence experience a de-con fi nementand therefore a reduction of the secular frequency. Qualita-tively this behavior can be seen in fi gure 3, even though thissimple picture neglects the motion of the shell ions while themotion of the core ions is excited. This is a reasonableapproximation as can be seen from fi gure 2 for an excitationclose to the single Be + ion secular frequency ( ) . One noticeable feature is that the secular spectra of indi-vidual Be + ions having symmetric positions with respect to thetrap center are not identical. The reason for these differences isthat the ion crystal is not exactly symmetric when fl ipping the z -axis. Depending on the exact con fi guration of their sur-roundings, ions at symmetric positions experience differentlocal potentials. It should be noted that the asymmetry is notdue to the random initial conditions of the simulation. Themost stable ion con fi guration does not necessarily exhibitmirror symmetry with respect to the xy -plane. + shell ions Next we investigate the motion of the 80 Mg + shell ions ofthe ion crystal in fi gure 1 ( b ) in order to understand theresonance splitting near the single Mg + resonance into twopeaks around 276 kHz and around 408 kHz ( see fi gure 2 ( b )) .The excursions along the x -axis of these ions in the timedomain are shown in fi gures 4 ( a ) and ( b ) for these two fre-quencies as obtained from our MD simulation. At 276 kHz itappears that ions with smaller magnitude of x i á ñ experience alarger excursion than those with larger magnitude of x i á ñ . Forthe peak at 408 kHz it is the other way around. The situationcan be clearly seen in fi gures 4 ( c ) and ( d ) , where the corre-lation between the motional amplitude and averaged position x i á ñ is shown. Figures 4 ( e ) and ( f ) sketch the motion of the Figure 2.
The secular spectrum calculated by the moleculardynamics simulation. ( a ) The light red and the dark blue curves referto the secular spectrum of a single trapped Be + ion and a singletrapped Mg + ion respectively, centered at the vertical dashed lines. ( b ) Secular spectrum of all Be + and all Mg + ions combined withinthe mixed Coulomb crystal shown in fi gure 1 ( b ) . The motion of thetwo species is coupled due to the mutual Coulomb interactions.Therefore, the resonances of the Mg + ions also show up in thespectrum of the Be + ions and vice versa. Figure 3.
Secular spectra of the individual Be + ( core ) ions ( in darkblue ) in the mixed Coulomb crystal of fi gure 1 ( b ) . The bottom plotshows the average that is also seen in fi gure 2 ( b ) . One might think that the ion shell does not generate a potential well on itsinside. This, however, is only true for an in fi nitely long cylinder withhomogeneous and symmetric charge distribution. ( )035002 B-m Ann
Secular spectra of the individual Be + ( core ) ions ( in darkblue ) in the mixed Coulomb crystal of fi gure 1 ( b ) . The bottom plotshows the average that is also seen in fi gure 2 ( b ) . One might think that the ion shell does not generate a potential well on itsinside. This, however, is only true for an in fi nitely long cylinder withhomogeneous and symmetric charge distribution. ( )035002 B-m Ann et al igure 4. ( a ) , ( b ) Trajectories of the shell ions projected on the x -axis when the frequency of the driving force is at either of the components ofthe split resonance at 276 kHz ( left ) or 408 kHz ( right ) ( see fi gure 2 ( b )) . The motional excitation is introduced at t =
0, and the oscillatorymotions of the ions are found to be stationary at t > μ s. ( c ) , ( d ) Correlation diagrams between the ion amplitudes and time averagedpositions x i á ñ . Gray lines in the diagrams are parabolic fi ts to guide the eye. ( e ) , ( f ) Axial view of the arrangement of the Be + core ions ( darkblue ) and the Mg + shell ions ( light red ) . The green arrows sketch the motional amplitude of the shell ions ( not to scale ) . The motion alsocouples to the core ions and makes them oscillate at the same frequencies ( not shown in the fi gure ) . ( g ) , ( h ) The trap potential for the shellions along x direction is depicted with the dashed curves. The solid curves indicate the modi fi ed potential due to the interaction with the coreions ( not to scale ) . ( )035002 B-m Ann
0, and the oscillatorymotions of the ions are found to be stationary at t > μ s. ( c ) , ( d ) Correlation diagrams between the ion amplitudes and time averagedpositions x i á ñ . Gray lines in the diagrams are parabolic fi ts to guide the eye. ( e ) , ( f ) Axial view of the arrangement of the Be + core ions ( darkblue ) and the Mg + shell ions ( light red ) . The green arrows sketch the motional amplitude of the shell ions ( not to scale ) . The motion alsocouples to the core ions and makes them oscillate at the same frequencies ( not shown in the fi gure ) . ( g ) , ( h ) The trap potential for the shellions along x direction is depicted with the dashed curves. The solid curves indicate the modi fi ed potential due to the interaction with the coreions ( not to scale ) . ( )035002 B-m Ann et al hell ions in the axial view represented by the lengths of thegreen horizontal arrows.For a simpli fi ed and intuitive picture, we again invokethe additional potential generated by the other group of ions.Figures 4 ( g ) and ( h ) show how the core ions ( dark blue ) modify the potential for the shell ions ( light red ) along thedirection of the excitation fi eld. The dashed curves refer to thetrap potential without the core ions, while the solid linessketch the total potential. As can be seen in fi gures 4 ( a ) and ( b ) , the shell ions close to x = x i á ñ have theirpotential enhanced with a corresponding up-shift of theirresonance. The situation is, however, less simple than for themodi fi cation of the core ions ’ potential by the shell ions. Thishas the following two reasons: The effect on ions withintermediate x i á ñ washes out the splitting of the resonance,which may be represented by the broadening of the down-shifted peak in fi gure 2 ( b ) . In addition, the core ions are not aswell fi xed in space as the shell ions are when driving theresonance of the core ions around 1 MHz.Since the modi fi ed potential due to the core ions is thereason for the resonance splitting, one expects the splitting toincrease with the number of core ions. This is indeed the caseas is shown in fi gure 5. The hand-waving arguments for the secular spectra of mixedcrystals are valid for large enough numbers of both ion spe-cies. The spatial distributions of the ions are not preciselyreproducible with every MD simulation run because the initialpositions are randomized. However, with a decent number ofboth ion species, the geometry of the crystal can be alwayscharacterized by a nested shell and core structure with radialsymmetry. As a result, the features described in section 2.2are also conserved in its secular spectra ( see appendix C fordetails ) .When the number of one species gets much larger thanthe other, such a simple geometric con fi guration does notalways show up, and the ion distribution is not regularlyreproduced for different initial conditions. For the extreme example with only one Mg + ion within a large number of Be + ions, the position of the Mg + ion in the Be + crystal turns outto be essentially random. The local potential which the Mg + ion experiences is dependent on its exact position and so is itssecular frequency. This is what we see in the experiment andalso in the simulations. As a consequence, it is dif fi cult toidentify a small number of impurity ions within a largercrystal by their secular frequencies. On the other hand, themajority ions reliably produce secular frequencies close totheir single ion resonance.
3. Experiments
The experimental setting is sketched in fi gure 1 ( a ) with theparameters provided in section 2. It is described in more detailin [ ] . The ion crystals are imaged with an objective havinga focal length of 76 mm at a distance of about 83 mm fromthe trap center and are observed with an electron multiplyingcharge-coupled device camera. Due to chromatic aberration,we cannot simultaneously image the Be + and Mg + ions.Instead we move the position of the objective to focus oneither of the ion species. An electrical motor stage isemployed for this purpose.To obtain fl uorescence from both ion species, we areaddressing both cooling transitions, the S P – ( naturallinewidth γ / π ≈
19 MHz ) and the S P – ( naturallinewidth γ / π ≈
42 MHz ) for Be + and Mg + respectively.The cooling lasers at 313 and 280 nm are both detuned bymore than 100 MHz. They run axially in opposite directionsthrough the ion trap with intensities of ≈
300 W m − and ≈ kW m – for Mg + and Be + respectively, determined fromthe laser powers and beam diameters. We adopted theserelatively low intensities in order to reduce distortions of thecrystal structure due to radiation pressure.In addition to the rods forming the quadrupole, there arefour outer rod electrodes shown with dashed circles in fi gure 1 ( a ) ( right ) . Three of these electrodes are utilized tocompensate for stray electric fi elds near the trap center. Theexcitation force f ex is created by applying an ac voltage atfrequency ω to the fourth electrode. Laser cooling is suf fi -ciently effective, such that the ion crystals stay intact whenthe secular vibrations are excited. A mixed Coulomb crystal is presented in fi gure 6, where theimages of Mg + and Be + are obtained separately by refocusingthe imaging system. Due to its strong chromatic aberration,we observe only a small almost homogeneous backgroundfrom one ion species when we focus on the other. Therefore,the data was taken with both cooling beams on.In contrast to our simulations, radiation pressure leads tosomewhat asymmetric crystals. This effect could have beenavoided by detuning the cooling lasers even further or byreducing their intensity. However, this would require stronger Figure 5.
The frequency separation between the two split resonancesof the shell ions as a function of the number of core ions obtainedfrom molecular dynamics simulations. The number of shell ions is fi xed to 80. The insets show the corresponding crystal structure. ( )035002 B-m Ann
The frequency separation between the two split resonancesof the shell ions as a function of the number of core ions obtainedfrom molecular dynamics simulations. The number of shell ions is fi xed to 80. The insets show the corresponding crystal structure. ( )035002 B-m Ann et al ecular excitation due to the reduced fl uorescence and hencemight eject the ions from the trap. Nonetheless, we believethat it is still reasonable to compare experiment with simu-lations. By comparing the shape and size of the ion crystalwith the simulation, we roughly estimate the number of ionsin fi gure 6 to be 500 for Mg + and 500 for Be + [
27, 35 ] .Several methods have been demonstrated to estimate themotional amplitude of the ions during secular excitation [ ] .In this work we measure the increase of the fl uorescence ofthe ions that is obtained with a largely red detuned coolinglaser due to the periodic Doppler shift closer to resonance.The secular spectrum is obtained by scanning the excitationfrequency ω . The excitation voltages were carefully optimizedto be not too large to disturb the crystal structure, and not toosmall to observe a suf fi cient amount of change in fl uores-cence. They ended up to be between 0.1 and 1.5 V. Based on fi nite element modeling of our geometry, a voltage modula-tion of 1 V at one of the outer rod electrodes induces anelectric fi eld of 0.75 V m – at the trap center. This leads to anexcitation force that is an order of magnitude stronger than inthe simulations. According to our MD simulations, strongerexcitation force should mainly in fl uence the cooling time,rather than the secular spectrum of the crystal that is formed.Figure 7 shows an experimental secular motion spectrum thatwas recorded for a mixed crystal that roughly corresponds insize and composition to the mixed crystal shown in fi gure 6.As in the simulations, the resonances of the mixedcrystals are signi fi cantly broadened in comparison with thepure ion crystals. In addition, we fi nd the spectrum near thepure Mg + ion crystal resonance to be split into two compo-nents, just as in the simulations and the simpli fi ed modeldescribed above. The observed splitting of 210 kHz is consistent with the simulation performed with a similarnumber ratio between Mg + and Be + ions, considering theuncertainty in the estimation of the number of Mg + and Be + ions. We also con fi rm the broadened resonance near the pureBe + crystal resonance. The down-shift of this resonance is notpredicted by our MD simulations but could be due to a sig-ni fi cantly stronger motional excitation in the experiment thanin the simulations. The in fl uence of the excitation strength onthe secular frequencies has been discussed in [ ] .It should be pointed out that the images shown in fi gure 6are time averaged and do not necessarily re fl ect the instan-taneous positions of the ions. At fi nite temperature, ions mayexchange their positions from one site to another. We expectthe hopping have a minor effect on the secular spectrumshown in fi gure 7 under our experimental conditions.
4. Conclusion
We have investigated the motional resonances of dual-speciesCoulomb crystals, both theoretically and experimentally.Motional coupling between the two species gives rise to non-trivial characteristics in the motional resonances that do notoccur in single-species Coulomb crystals. Both in the MDsimulation and in the experiment, we observed that theresonance of the lighter ion species is broadened and shiftedto higher frequencies, while the resonance of the heavierspecies is split into two components. Our fi ndings show thatthis triple peak structure appears in the secular spectrum ofmixed ion crystals even when only two ion species areinvolved. This could be misinterpreted as an appearance of athird ion species. Our simple model of interactions between Figure 6.
Images of a mixed ion crystal comprised of about 500 Mg + ions and about 500 Be + ions recorded with an electron multiplyingCCD ( EMCCD ) camera. The fl uorescence images of the two ionspecies are obtained and shown separately, even though they arefrom the same ion crystal. The asymmetric geometry is attributed toradiation pressure from the cooling beams. Stray potentials and othertrap imperfections may also cause such an asymmetry. Figure 7.
Experimental secular spectrum of a Mg + and Be + ioncrystal detected by measuring the fl uorescence of the correspondingion species. ( a ) Pure Mg + ( light red ) and pure Be + ( dark blue ) crystals. ( b ) Two independently prepared mixed crystals, both ofwhich similar to the one presented in fi gure 6. The light red ( left ) anddark blue ( right ) curves are secular spectra detected by observing the fl uorescence of the Mg + and Be + ions respectively. Therefore, theamplitudes of the left and right parts do not compare and arerescaled. The excitation was done with an amplitude ( peak-to-peak ) of 0.22 V and 0.24 V for the pure Mg + and Be + crystals and with1.5 V and 1.2 V for the mixed crystals respectively. ( )035002 B-m Ann
Experimental secular spectrum of a Mg + and Be + ioncrystal detected by measuring the fl uorescence of the correspondingion species. ( a ) Pure Mg + ( light red ) and pure Be + ( dark blue ) crystals. ( b ) Two independently prepared mixed crystals, both ofwhich similar to the one presented in fi gure 6. The light red ( left ) anddark blue ( right ) curves are secular spectra detected by observing the fl uorescence of the Mg + and Be + ions respectively. Therefore, theamplitudes of the left and right parts do not compare and arerescaled. The excitation was done with an amplitude ( peak-to-peak ) of 0.22 V and 0.24 V for the pure Mg + and Be + crystals and with1.5 V and 1.2 V for the mixed crystals respectively. ( )035002 B-m Ann et al ore and shell ions qualitatively explains the observation andcan help to properly interpret the secular spectrum of mixedion crystals. Acknowledgments
Byoung-moo Ann acknowledges fi nancial support fromDAAD ( German academic exchange service ) foundation.This project has received funding from the EuropeanResearch Council ( ERC ) under the European Union ’ s Hor-izon 2020 research and innovation programme ( grant agree-ment No. 742247 ) . We also thank Prof Hans A Schuessler forreading the manuscript and giving valuable suggestions.
Appendix A. Effect of micromotion
Throughout this study, we have utilized the time-averagedeffective potential ( equation ( )) in the MD simulation ratherthan the full time-dependent potential of the RF trap. Themicromotion induced by the RF potential could affect thespace charge distribution of the coolant or the sympatheticallycooled ions which may change the mechanical couplingbetween them. We expect that such an effect is negligiblebecause the time scale of the secular motion is an order ofmagnitude slower than that of the micromotion. In addition,the amplitude of the micromotion is smaller than the typicalion-to-ion distance. To validate the use of the time-averagedpotential, we simulated the secular spectra of a mixed ioncrystal including the full time-dependent potential. The resultis shown in fi gure A1. The dots in fi gure A1 refer to thesecular spectra of an ion crystal comprised of 80 Mg + ionsand 20 Be + ions under the full time-dependent potential,whereas the solid lines refer to the secular spectra given bythe same ion crystal, but under the effective potential. Thenumber of samples for the full time-dependent potential caseis reduced compared to the effective potential case because ofthe much longer computation time required for the simulation.The full time-dependent and the effective potential simulation result in almost identical spectra. This con fi rms that our dis-cussion on the motional coupling is still valid under thein fl uence of micromotion. Appendix B. Ion mass dependence
In our simulations we have fi xed the mass of the ion speciesto re fl ect the experimental conditions. However, we believethat the qualitative features of the motional spectra are quitegeneral and will be reproduced when changing parameterssuch as ion species, number ratios, and trap depths. As anexample, in this section we show how the mass of the lighter ( core ) ions affects the motional spectrum of the shell ions ( section 2.4 ) . We simulated the secular spectra of ion crystalsconsisting of 80 Mg + ions and 30 core ions for core ionmasses between 3 and 15 u. The resulting spectra of the Mg + ions are shown in fi gure B1. While there are sig-ni fi cant differences between the spectra, the splitting of thesingle-ion resonance is clearly visible in all of them. Appendix C. Randomized ion distribution
When running the MD simulations, the distribution of ionswithin the resulting mixed crystals depends on the ions ’ initialpositions. Therefore, the precise shapes of the secular spectraare also dependent on the initial conditions. The deviationsare more prominent with a weaker cooling force. In this casethe resonance widths become narrower so that more detailsappear in the secular spectrum. Even then, the prominentfeatures in the spectrum discussed in this work are alwaysreproduced. To illustrate this, we have performed 10 MDsimulations with 80 Mg + and 20 Be + ions under identicaltrapping conditions, but with different ( random ) initial con-ditions. We have used the same parameters given in the maintext, except that the driving force as well as the cooling is 10times weaker in order to enhance the differences. Obtainedsecular motion spectra are presented in fi gure C1. Some of the fi ne structures in the spectra depend on the initial condition as Figure A1.
A comparison of secular spectra obtained using the time-averaged secular potential ( solid lines ) and the full time-dependentpotential ( dots ) . The secular spectra of the two cases agree well witheach other. Figure B1.
Motional spectra of the Mg + shell ions for differentcore ion masses. The number of shell ions is fi xed to 80, and thenumber of core ions to 30. ( )035002 B-m Ann
Motional spectra of the Mg + shell ions for differentcore ion masses. The number of shell ions is fi xed to 80, and thenumber of core ions to 30. ( )035002 B-m Ann et al xpected. Despite of that, all spectra show a broad and a splitresonance and closely resemble fi gure 2 ( b ) . ORCID iDs
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10 secular spectra of a mixed Coulomb crystal comprisedof 80 Mg + ions and 20 Be + ions obtained with molecular dynamicssimulations under identical trapping and cooling conditions, but withdifferent initial conditions. ( a ) Spectra averaged over all Mg + ions. ( b ) Averaged over all Be + ions. To obtain sharper resonances inorder to see fi ner details, the drag force was reduced to α = × − kg s – here. Without this, all the spectra of this fi gure would essentially look identical to fi gure 2 ( b ) . ( )035002 B-m Ann
10 secular spectra of a mixed Coulomb crystal comprisedof 80 Mg + ions and 20 Be + ions obtained with molecular dynamicssimulations under identical trapping and cooling conditions, but withdifferent initial conditions. ( a ) Spectra averaged over all Mg + ions. ( b ) Averaged over all Be + ions. To obtain sharper resonances inorder to see fi ner details, the drag force was reduced to α = × − kg s – here. Without this, all the spectra of this fi gure would essentially look identical to fi gure 2 ( b ) . ( )035002 B-m Ann et al ] Wübbena J B, Amairi S, Mandel O and Schmidt P O 2012Sympathetic cooling of mixed-species two-ion crystals forprecision spectroscopy
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